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Remark 2.4.8.
Let (A, μ, Δ) again be a bialgebra. Then the category comod-A of right A-comodules is a
tensor category as well. Given two comodules (M, Δ M ) and (N, Δ N ), the coaction on the tensor
product M ⊗ N is defined using the multiplication:
Δ M⊗N :
M ⊗ N Δ M ⊗Δ
−→
N
id
M ⊗ A ⊗ N ⊗ A
M ⊗τ⊗id
−→
A
M ⊗ N ⊗ A ⊗ A
id M⊗N ⊗μ
−→ M ⊗ N ⊗ A .
It is straightforward to dualize all statements we made earlier.
In particular, the tensor unit is the trivial comodule which is the ground field K with a
coaction that is given by the unit η : K → A:
K
η
−→ A ∼ = K ⊗ A .
Again, the associativity and unit constraints of comodules are inherited from the constraints
for vector spaces:
(M ⊗ N) ⊗ P ∼ = M ⊗ (N ⊗ P )
K ⊗ M ∼ = M ∼ = M ⊗ K .
2.5 Hopf algebras
Observation 2.5.1.
Let (A, μ) be a unital algebra and (C, Δ) a counital coalgebra over the same field K. We can
then define on the K-vector space of linear maps Hom(C, A) a product, called convolution. For
f, g ∈ Hom(C, A) this is the morphism
f ∗ g : C
−→ Δ
C ⊗ C −→ f⊗g
A ⊗ A −→ μ
A .
This product is K-bilinear and associative. In Sweedler notation
(f ∗ g)(x) = f(x (1) ) ∙ g(x (2) ) .
The linear map
C
−→ ɛ
K −→ η
A
is a unit for this product.
This endows in particular the space End K (A) of endomorphisms of a bialgebra A with the
structure of a unital associative K-algebra. It is, however, not clear whether in this case the
identity id A ∈ End K (A) is an invertible element of the convolution algebra.
Definition 2.5.2
We say that a bialgebra (H, μ, Δ) is a Hopf algebra, if, under the convolution product, the
identity id H has a two-sided inverse S : H → H, called the antipode.
Remarks 2.5.3.
1. The defining identity of the antipode
S ∗ id H = id H ∗ S = ηɛ
reads in graphical notation
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